MathDB

k = f(x)*(x+1)^n + g(x)*(x^n + 1)

Source: IMO Shortlist 1996, A6

8/9/2008

Let

n

be an even positive integer. Prove that there exists a positive inter

k

such that k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1) for some polynomials

f(x), g(x)

having integer coefficients. If

k_0

denotes the least such

k,

determine

k_0

as a function of

n,

i.e. show that k_0 \equal{} 2^q where

q

is the odd integer determined by n \equal{} q \cdot 2^r, r \in \mathbb{N}. Note: This is variant A6' of the three variants given for this problem.

algebrapolynomialfunctionIMO Shortlist

First rectangle can be placed within the second one

Source: IMO Shortlist 1996, G6

8/9/2008

Let the sides of two rectangles be

\{a,b\}

and

\{c,d\},

respectively, with

a < c \leq d < b

and

ab < cd.

Prove that the first rectangle can be placed within the second one if and only if \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.

geometryrectanglerotationIMO ShortlistInequalityfour variables

Finite number of coins placed on an infinite row of squares

Source: IMO Shortlist 1996, C6

8/9/2008

A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.

algorithminvariantcombinatoricsIMO Shortlistchip-firing

6

Problems(3)